I don’t know about degrees in Folk Music, but you can get a degree in Ethnomusicology, which is the study of the social and cultural aspects of music and dance in local and global contexts. This typically includes the study of specific cultural styles of music include folk music (as well as things like Jazz).
I do think some schools have Folk Music degrees, Newcastle University in the UK I think had the first such program.
I grew up with folk music and play several instruments that fall under folk, including guitar, folk harp, bodhrán, various flutes, fife, tin whistle and recorder. The winds I learned in school. We learned a lot of local folk music in elementary school. Guitar and harp I took lessons, and although I didn’t specifically study folk on either I learned to easily use what I learned in the folk context. Bodhrán I am self taught.